11924 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 22, No 2, 2024, pp. 275 - 292 https://doi.org/10.22190/FUME230820049I © 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper CARBON NANOTUBE UNDER PULSED PRESSURE Marat A. Ilgamov1,2,3, Aigul A. Aitbaeva3, Igor S. Pavlov4,5, Sergey V. Dmitriev6,7 1Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow, Russia 2Institute of Mechanics and Engineering, KSC, Russian Academy of Sciences, Kazan, Russia 3Institute of Mechanics, UFRC, Russian Academy of Sciences, Ufa, Russia 4Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia 5Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia 6Institute of Molecule and Crystal Physics, UFRC, Russian Academy of Sciences, Ufa, Russia 7Department of Equipment and Technologies of Welding and Control, Ufa State Petroleum Technological University, Ufa, Russia ORCID iDs: Marat A. Ilgamov https://orcid.org/0000-0002-8766-8964 Aigul A. Aitbaeva https://orcid.org/0000-0002-4735-1671 Igor S. Pavlov https://orcid.org/0000-0002-2666-4943 Sergey V. Dmitriev https://orcid.org/0000-0002-6744-4445 Abstract. The radial dynamics of a single-walled zigzag carbon nanotube under pulsed pressure is studied. Uniform external pressure is applied instantly, then remains constant for a certain time, and then is instantly released. This loading scheme allows one to consider a carbon nanotube under plane strain conditions and replace it with a circular ring formed by one zigzag row of carbon atoms. The bending deformation of the ring in its plane is described by an equation based on the Kirchhoffs hypothesis. An effective parameter is used, including the bending stiffness of the ring and areal density. The regimes of oscillatory motion and exponential growth of radial displacements are investigated depending on the magnitude and duration of the applied pressure. Alternatively, the ring is analyzed in terms of a molecular dynamics model with a reduced number of degrees of freedom, taking into account the plane strain conditions. With the help of molecular dynamics, the limits of the thin shell theory are established. For (n, 0) CNTs with n > 15, in the regime of small-amplitude vibrations, the discrepancy between the continuum model and molecular dynamics calculations does not exceed 5% and decreases with increasing n. Key words: Carbon nanotube, Molecular dynamics, Shell theory, Pulsed pressure, Natural frequencies Received: August 20, 2023 / Accepted December 05, 2023 Corresponding author: Sergey V. Dmitriev Institute of Molecule and Crystal Physics, UFRC RAS, Oktyabrya Ave. 71, 450054 Ufa, Russia E-mail: dmitriev.sergey.v@gmail.com https://orcid.org/0000-0002-8766-8964 https://orcid.org/0000-0002-4735-1671 https://orcid.org/0000-0002-2666-4943 https://orcid.org/0000-0002-6744-4445 276 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV 1. INTRODUCTION Free and forced vibrations of carbon nanotubes (CNTs) have been analyzed in a number of studies [1-13] because they affect the operation of nano-devices such as sensors, actuators, etc. The physical and mechanical characteristics of CNTs, including their behavior under static and dynamic loading, are often investigated using the methods of molecular dynamics and continuum mechanics [1-5]. In the latter case, the effective dimensions of CNTs, effective elastic and mass parameters are introduced. In this work, the molecular dynamics and the theory of thin shells with a parameter determined from the comparison of natural frequencies within the framework of these models are used. The effect of a uniform dynamical pressure on a single-walled CNT is considered. The effect of pressures on CNTs is discussed in reviews [6, 7]. There are experimental studies of the dynamic, structural and electrical properties of single-walled and multi-walled CNTs, depending on the uniform static and dynamic pressure [8-13]. These works also consider theoretical modeling within the framework of molecular dynamics and continuum mechanics. The deformation of isolated CNTs under static pressure is analyzed, which is a necessary step in studying the complex behavior of a CNT bundle under dynamic pressure. It has been shown [10, 11] that with an increase in static pressure up to p1≈3D/R, the CNT cross-section acquires the shape of an ellipse (here D is the CNT wall bending stiffness and R is the CNT radius). In the elliptic state, the radial stiffness of the CNT decreases by two orders of magnitude [11]. Further increasing the pressure up to p2≈p1[1-ln(F2/F1)] results in a cross-sectional shape with two opposite points of zero curvature. With a further increase in pressure up to p3≈p1[1-ln(F3/F1)], the deformation of the cross-section increases and the electrical properties of the CNT change (transition to semiconductor state takes place). Here F1, F2 and F3 are the cross-sectional areas corresponding to the pressures p1, p2 and p3. For a single-walled armchair CNT (10, 10), p1=1.55, p2=1.75 and p3=2.2 GPa [10]. The static stability and deformations of thin elastic circular rings under nonuniform pressures p=p0(1+qcos) were studied and it was found that the ring deforms only in a doubly symmetric fashion, while static buckling into an asymmetric pattern was not observed [14]. Vibrational properties of long hollow cylindrical shells interacting with the external medium were analyzed in the works [15, 16]. The structural transformations in CNT bundles during laser compression and the resulting oscillations have been considered [8, 9, 12]. In these and other works [17, 18], Raman spectroscopy is widely used in experimental studies. The behavior of CNTs was studied experimentally at different distances between them in the bundle (from 1.5 to 1.7 nm). In [13], the change in the properties of a single multilayer CNT under shock compression through the surrounding elastic material was studied. The radial breathing mode frequency of various zigzag and armchair CNTs with radii between 3.5 and 8.1 Å have been determined using the first-principles calculations [19]. Dispersion relations for long-wavelength phonons in graphite and achiral single-walled CNTs have been analyzed using a continuum model [20]. Analysis of vibration behavior of carbon nanomaterials is important for design of nano-mass, nano-force and strain sensors [21-29] as well as microelectromechanical systems for various purposes [7, 30]. Dynamic loading of carbon nanomaterials is considered in solving important problems related to the damping of shocks and vibrations as well as projectile protection. Molecular dynamics simulations and experiments were conducted to study energy dissipation in CNT Carbon Nanotube under Pulsed Pressure 277 films under ballistic impact [31, 32, 33]. Energy dissipation by collapsing CNTs in the bundle under shock loading was analyzed in the work [34]. In this work, we analyze the response of single-walled CNTs to a uniform dynamic pressure within the framework of the theory of thin shells and molecular dynamics. 2. PROBLEM STATEMENT It is assumed that a time-varying external uniform excess pressure p(t) is applied at time t=0 to a single-walled zigzag CNT, which at t<0 is under the same external and internal pressure p0 (for example, atmospheric pressure). For a uniform pressure, zigzag CNT can be considered as a ring formed by single zigzag chain of carbon atoms (see Fig. 1). As the length of the valence bond, we take l=1.273Å which is slightly less than the experimental value in graphene, 1.4 Å, due to the effect of curvature of the CNT wall, especially for CNTs with a small radius. In the ring, the distance between the carbon atoms, projected onto the xy plane, is equal to a=lcos30º=1.102 Å. Fig. 1 (a) The structure of a zigzag CNT wall and (b) representation of the zigzag CNT by the chain model. The valence bond length is l=1.273Å, the distance a=lcos30º=1.102 Å, CNT radius is R. Cross section of the zigzag CNT (n,0) is presented by N=2n carbon atoms numbered with the index i=1,…,N; the case of N=30 is shown in (b). A ring of width b=3l/2=1.91 Å is considered with periodic boundary conditions along the z axis. The area per one C atom S=2.105 Å2 is shown by dotted lines (equilateral triangle). A continuum mechanics model considers a ring formed by two cross sections at a distance b=3l/2=1.91 Å symmetrically relative to a zigzag row of carbon atoms. The radius R is determined from the equality 2 sina R N  = , (1) where N is the number of atoms forming the ring. Note that N=2n for the zigzag CNT (n,0) The area per atom is equal to S=2.105 Å2. The mass of a carbon atom is M=1.99×10-26 kg or M=12 a.m.u. Here we use angstroms, picoseconds, and electronvolt as units of measure for distance, time, and energy, respectively. In these units the carbon atom mass is M=1.244×10-3 eV·ps2/Å2. The effective area density is 278 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV 2 2 4 eV ps 5.908 10 Å M h S − = =  , (2) where  and h are the effective bulk density and ring thickness. 2. ELASTIC RING MODEL Before pressure p(t) is applied, a perfectly circular ring performs small-amplitude free oscillations in its plane. These free vibrations will be represented as a sum of a few first harmonics. It is assumed that the pressure begins to act at the time of the greatest deviation for each harmonic. Then the initial conditions with respect to the deflection function w(,t), where  is the central angle, are taken in the form 0 2 0 0 2 2 ( ,0) cos( ), ( ,0) 0, ( 1) , 0, m m m m w W m w W W m = − =  =   = = −     (3) where m is the harmonic number, W2 0 is the initial amplitude of the harmonic m=2. The overdot denotes the derivative with respect to time t. The uniform distribution of the amplitudes Wm 0 corresponds to =0 and for >0 the amplitude decreases with the harmonic number. The linear equation of motion of the ring in its plane with respect to the deflection function w(,t) has the form [14]: 6 4 2 4 2 4 2 6 4 2 4 2 2 2 3 * 2 2* 3 2 2 2 3 0, 3 , , , . 12(1 ) w w w w w R w w p D D Eh p D hp R          + + +  + + − =               = =  = =  −  (4) Here, D is the effective bending stiffness, E and  are the effective modulus of elasticity and Poisson’s ratio, respectively, p* 2 is the critical pressure at which the loss of stability of the circular shape of the ring with respect to the lowest harmonic m=2 occurs (it is indicated as p1 above [10]). Since the width of the ring b enters the values of masses, stiffness and pressure, it is canceled in Eq. (4). Eq. (4) does not take into account the influence of the surrounding gas environment, which is characterized by the parameter f R/(h), where f is the density of the surrounding medium,  is a complex function of the input quantities [14, 15]. The real and imaginary parts of  are responsible for the added mass of the medium and the radiation of energy from the tube. Except in special cases not covered here,  does not increase the value of this parameter. Therefore, the influence of the external environment on the motion of the tube can be judged by the dimensionless parameter f R/(h). If its value is much less than one, then the influence of the environment is small. Taking into account the above value of h and the equality 2R≈aN, this condition can be written as f R~10-6 kg/m2 or f N=4×104 kg/m3. For air at atmospheric pressure f =1.2 kg/m3, and this condition gives R~103 nm or N=3×104. We will consider the dynamics of a ring formed from atoms of Carbon Nanotube under Pulsed Pressure 279 order 10 to 100. Therefore, we will not take into account the influence of the external environment. With a slow increase in pressure on the tube, the considered influence of the medium on the tube is absent. 3. MOLECULAR DYNAMICS MODEL CNT under plane strain conditions can be simulated with the use of the chain model proposed in [35, 36] and adapted for simulation of CNTs in [37, 38]. The chain model considers one zigzag row of carbon atoms moving in the xy-plane as shown in Fig. 1(b). The atoms are numbered with the index i=1,…,N; their positions are defined by the radius-vectors ri. The Hamiltonian of the chain is 2 1 | | 2 N i B A i M H r U U = = + + , (5) where the first term gives the kinetic energy of the chain and the terms UB and UA describe the potential energy of valence bonds and valence angles, respectively. The energy of valence bonds is  = + −−  = N i iiB arrU 1 2 1 )|(| 2 , (6) where  =25.279 eV/Å2 is the valence bond stiffness. The energy of valence angles is  = += N i iAU 1 ]cos1[ , |||| ),( cos 11 11 iiii iiii i rrrr rrrr −− −− = +− +− , (7) where  =0.00166 eV is the valence angle stiffness. External forces applied to each atom were added to the equations of motion following from the Hamiltonian Eq. (5) to simulate a uniform conservative external pressure p(t). The chain model was successfully used for simulation of carbon nanostructures, in particular, structure and thermomechanical properties of CNT bundles were analyzed in the works [37, 40-45], folded and scrolled packings of carbon nanoribbons in [46], crumpled graphene under compression in [47]. Chain model can also be applied to the analysis of other two-dimensional nanomaterials [48, 49]. For N carbon atoms of the CNT cross section numbered by the index i (see Fig. 1(b)) the radial displacements can be presented as the sum of N/2+1 Fourier harmonics 2 1 0 2 1 1 1 2 2 2 cos( ) cos sin N i N m m m mi mi w f f i f g N N N N N − =        = +  + +           , (8) where the coefficients of the expansion are calculated as follows 280 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV 1 0 1 0 2 cos , 0,..., 2, 2 sin , 1,..., 2 1. N m i i N m i i mi f w m N N mi g w m N N − = − =   = =      = = −      (9) The following initial radial displacements are applied 4 0 2 2 cos , 0. i m m i mi w W N w =   =     =  (10) It can be seen that initially only three first bending harmonics shown in Fig. 2 are excited (the harmonics m=0 and m=1 are tensile and translational, respectively). We take equal amplitudes for the three harmonics, Wm 0=10-3R, m=2,3,4, which is an analog of Eq. (3) for =0. Fig. 2 The three first radial vibrational modes of the CNT cross section Table 1 CNT radius R (in Å) and numerical values of  (in Å2/ps) for various N and m=2,3 and 4. N R  (m=2)  (m=3)  (m=4) 18 3.179 54.75 52.84 52.71 30 5.278 56.71 56.03 54.78 42 7.382 57.55 57.09 56.45 54 9.487 57.84 57.55 57.18 66 11.59 57.98 57.78 57.49 78 13.70 58.06 57.92 57.74 90 15.80 58.12 58.01 57.86 102 17.91 58.15 58.07 57.95 114 20.02 58.18 58.11 58.02 126 22.12 58.19 58.14 58.07 138 24.23 58.21 58.17 58.11 150 26.33 58.22 58.19 58.14 Carbon Nanotube under Pulsed Pressure 281 The amplitudes of N/2+1 Fourier harmonics are 0 0 2 2 2 2 | |, , 1,..., 2 1, | | . m m m N N W f W f g m N W f = = + = − = , (11) Numerical examples will be given for the CNT with the number of atoms N=200. For such a large value of N we take =58.2 Å2/ps, see Table 1. From Eq. (1) one finds the CNT radius R=aN/(2)=35.08 Å and from Eqs. (2) and (4) D=2.001 eV. The equations of motion of the atoms representing the CNT cross section are integrated numerically by the symplectic Stormer method of the sixth order of accuracy [39]. Time evolution of the Fourier harmonic amplitudes, Wm(t), is analyzed. 4. NATURAL VIBRATION FREQUENCIES OF CNT The theory of thin elastic shells gives the following expression for the natural circular frequencies of bending vibrations of a cylindrical shell [14] 2 2 2 ( 1) 1 m m m R m −   = + , (12) where m≥2 is the harmonic number and R is the shell radius. The numerical values of the parameter  included in Eqs. (4) and (12) were determined in [40, 41] by comparing the frequencies m by Eq. (12) and by the chain model; they are given in Tab. 1. The analytical approximation (N,m) given in [41] gives a satisfactory result only for the mode m=2. Here, this approximation is somewhat improved and has the form 5 2( , ) ( )N m A sm q N− = − − , (13) where A=58.2 Å2/ps, s=7.2×103 Å2/ps, q=8.4×103 Å2/ps. For N>150 and m≤4 the value =58.2 Å2/ps remains constant. In Fig. 3, the dependencies (N,m) according to Eq. (13) are given for m=2,3 and 4 together with the numerical data (scattered) obtained with the help of the chain model. The value of  increases with an increase in the number of atoms N and decreases with an increase in the harmonic number m. The ratio of the wavelength around the circumference of the ring 2R/m ≈ aN/m to the thickness h is aN/(mh). Taking into account the value of a=1.102 Å and the value of h=3.3 Å, which is the interlayer distance in graphite, we obtain this ratio equal to N/(3m). Equation of motion Eq. (4), based on the Kirchhoff hypotheses [14], is valid for N/(3m)>10. It was obtained without taking into account the inertia of rotation of the ring element and transverse shear, which affect the solution for N/(3m)<10. In this case, it is necessary to use the Timoshenko model, which takes these factors into account [50, 51]. On the other hand, at N/(3m)<10, there are less than 15 atoms per half-wavelength, and the inaccuracy of the equations of continuum mechanics becomes noticeable. Taking into account the inertia of rotation and transverse shear leads to a decrease in the natural vibration 282 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV frequencies in comparison with the result based on Kirchhoff hypotheses. This trend is seen in Fig. 3. Fig. 3 Dependence of the parameter  entering Eq. (13) on the number of atoms N forming a single-layer ring and the harmonic number m of radial vibrations. The dots show the numerical values from [18]. 5. LONG-TERM PRESSURE RESPONSE At t=0 the external pressure p=const is instantly applied to the CNT and acts for a long time. By a long duration of a pressure pulse of constant intensity, we mean a time interval t<, an order of magnitude or more exceeding the oscillation period 2/m. 5.1. Analytical Treatment Looking for a solution to equation Eq. (4) in the form 2 ( )cos( )m m w W t m = =  , (14) from Eq. (4) and Eq. (14) we obtain 02 =+ mmm WW , (15) were 2 2 2 3 1 1 m m m    =  −  −  , (16) Here, m is the natural frequency of the m-th harmonic at zero pressure defined by Eq. (12), m is the frequency under the action of excess pressure p. The critical pressure value for the CNT with N=200, R=35.08 Å, and D=2.001 eV, with respect to the harmonic m=2, in accordance with Eq. (4), is Carbon Nanotube under Pulsed Pressure 283 3 * 4 2 3 3 eV 1.391 10 22.28MPa Å D p R −= =  = . (17) From the condition m =0 one finds from Eq. (16) the critical values of pressure for other harmonics, 2 * * 1 , 3 m m m p p − = (18) and hence, for the considered CNT, * * 4 3 2 8 3.709 10 59.41MPa 3 p p −= =  = (19) * * 4 4 25 6.955 10 111.4MPap p −= =  = (20) Next, we analyze the CNT dynamics after an instantaneously applied constant external pressure p. In the case of an abrupt increase in pressure at time t=0 from 0 up to a constant value p, the solution to Eq. (15) has the form ( ) m mt t m m mW t A e B e  − = + . (21) Satisfying the initial conditions Eq. (3), we obtain for the m-th harmonic 2 0 2 cos( ), 3 1,( ) cosh( ), 3 1. mm m m t mW t W t m     − =     − (22) Thus, depending on the ratio of the effective pressure to the critical pressure  and the number of the harmonic m, oscillations or/and an exponential growth of the initial deflections can be excited after the load is applied. For a relatively small value of p78) and the same initial deviation amplitudes (=0) this condition gives 2 21 2 12 2 L R R Rm m m m    − + +    , (25) where ⌈·⌉ denotes the round up operation. The mL value is somewhat less than the mR value. Therefore, the fastest increase in the amplitude is observed for the harmonic at the border of exponential growth and oscillatory motion. What has been said applies to the case of a uniform distribution of the initial deviation over harmonics (=0). At >0, the picture changes. At the beginning of the process, the harmonic with number m=2 prevails due to the nature of the initial conditions Eq. (3) and the presence of the factor (m-1)- in solution Eq. (24). With increasing time, the second term in Eq. (24), 2-cosh(3t) becomes larger than the first, cosh(2t), and so on. Thus, during dynamic deformation of a CNT with a decreasing distribution of small initial deflection over harmonics, a rearrangement in m occurs over time. At large time, the harmonic mL, which is determined approximately by Eq. (25), becomes dominant. In the presence of limiters of movement along the radius, for example, contacting media, the harmonic mL may not become dominant. Nonlinearities can also lead to this limitation. 5.2. Chain Model Results Let us first check the accuracy of Eq. (16) in predicting the main frequency of CNT vibrations at an external pressure p1, in particular, consider =2,3 and 7. For =2, p>p2 * and p2. From Eqs. (23) and (25) for =2 we find mR=⌊2.6⌋ and mL=⌈2.0⌉. Rounded values are mR=3 and mL=2. According to the above theory, in Eq. (24) the first sum is taken from 2 to 3, and the second - from 4. The harmonic m=2 grows exponentially with time and is the fastest growing harmonic. The remaining harmonics are periodic in time. In Fig. 5(a), the time dependence of the harmonic amplitudes is presented. In line with the theoretical prediction, W2 grows exponentially, while W3 and W4 fluctuate over time. Carbon Nanotube under Pulsed Pressure 285 Fig. 4 (a) Amplitudes of Fourier harmonics as the functions of time found for CNT with N=200 under external pressure p=18.2 MPa (=p/p2 *=0.816). (b) Frequency of the harmonic m=2 normalized to the natural vibration frequency at p=0 as the function of the applied pressure normalized to the critical value p2 *. Red line shows the analytical solution Eq. (16) for the elastic ring and black squares the numerical results for the CNT. For =3, p>p3 * and p3. From Eqs. (23) and (25) for =3 we find mR=⌊3.4⌋ and mL=⌈2.4⌉. Rounded values are mR=3 and mL=3. As can be seen from Fig. 5(b), W2 and W3 grow exponentially, while W4 performs oscillations. As predicted by the theory, the fastest growth is demonstrated by W3. For =7, p>p4 * and p4. From Eqs. (23) and (25) for =7 we find mR=⌊4.7⌋ and mL=⌈3.4⌉. Rounded values are mR=4 and mL=4. In Eq. (24), the first sum is taken from 2 to 4, and the second from 5. The harmonics m=2,3 and 4 grow exponentially, with the fastest growing harmonic m=4. The numerical results are presented in Fig. 5 (c) to confirm the theoretical predictions. Starting from m=5, oscillations occur with increasing frequency and decreasing amplitude around the motion determined by the harmonics m=2,3 and 4. 286 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV Fig. 5 (a) Amplitudes of Fourier harmonics as the functions of time found for CNT with N=200 under external pressure p corresponding to (a) =2, (b) =3, and (c) =7. 6. REACTION TO A PRESSURE PULSE Below we analyze the dynamics of CNTs due to a relatively short rectangular pressure pulse. 6.1. Analytical Treatment In the case of the pressure pulse shown in Fig. 6, one can use solution Eq. (24) within 0≤t≤, where  is the pulse duration. For t > , when p=0, =0, n=n , the solution to Eq. (15) has the form ( ) cos( ) sin( )m m m m mW t A t B t=  +  . (26) The integration constants are determined from the conditions Carbon Nanotube under Pulsed Pressure 287 2 0 2 2 0 2 cos( ), 3 1,( ) cosh( ), 3 1, sin( ), 3 1,( ) sinh( ), 3 1. mm m m m mm m m m mW t W m mW t W m      − =      − −     − =       − (27) The first line in Wm(t) and ( )mW t corresponds to the case when the effective pressure p) is determined by the expressions     −+− −−− = ,0)],(sin[)sinh()](cos[)cosh( ,0)],(sin[)sin()](cos[)cos()( 2 2 0 mmmmmm mmmmmm m m tit tt W tW (28) where i is the imaginary unit and we have introduced the notation 2 2 3 1 1 m m   = − − (29) in accordance with Eq. (16), so that m 2=m 2m 2. For =0 in both solutions of Eq. (28), the motion is a continuation of the initial motion defined by Eq. (3), w=W0mcos(mt)cos(m), since there is no pressure drop yet. In the case p=0 (=0, m=±1) from the first solution we obtain the same function, and the second solution Eq. (15) has no meaning. In the general case, the motion in the interval 0 by Eq. (28). Thus, during the pressure action (0), oscillations for each harmonic occur with a higher frequency m than m. In the case 3p>p2 *(m2-1), within the pressure pulse (0, oscillations occur with the same frequencies m, but with increased amplitudes. This solution is valid within the formulation of the problem in the linear approximation. 288 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV Example. Let us consider in more detail the dynamics of the harmonic m=2 in the CNT with the number of atoms N=200. According to Eq. (12), 2=0.127 rad/ps. For p=18.2 MPa (=p/p2 *=0.816, 2=0.429) one has from Eq. (16) 2=0.054 rad/ps. Since p1. Let us take =1.1, then 2=(p/p2 *-1)1/2=0.316, 2=22=0.040 rad/ps. For t< the solution is given by Eq. (24) and for t> the solution is given by the second line of Eq. (28):     2 0 2 cosh(0.040 ), 0 , cosh(0.040 )cos 0.127( ) 0.316sinh(0.040 )sin 0.127( ) , ( ). t tW t t tW    =   −  −  −    (31) It is clear that for p>p2 * there is a strong dependence of the amplitude of oscillations after the termination of pressure pulse on its duration . 6.2. Chain Model Results The time dependence of the harmonic amplitudes W2, W3 and W4, are shown for the CNT with N=200 for the pulsed pressure in Fig. 7. In Fig. 7(a) p=18.2 MPa (=0.816) and pulse duration is equal to =100 ps. This result should be compared with the one shown in Fig. 4(a), where the same pressure is constantly applied. Under the action of a pressure pulse, the CNT wall performs periodic oscillations, and the fundamental bending harmonic m=2 has a frequency 2=0.054 rad/ps. After the external pressure is removed, the amplitude of oscillations of the m=2 harmonic decreases somewhat, and the frequency increases in accordance with the theoretical prediction, see the first line of Eq. (28). Removing the pressure has little effect on the dynamics of higher harmonics m=3 and 4. In panels (b) and (c) of Fig. 7, the external pressure is greater than p2 *, namely, it corresponds to =2 and 3, respectively. The duration of the pressure pulse is =25 and 20 ps, respectively. These graphs should be compared with Fig. 5(a) and (b) respectively, where the same pressure is constantly applied. It can be seen from Fig. 7 that after the removal of external pressure, the exponential growth of W2 in (b) and W2 and W3 in (c) turns into a periodic motion with an amplitude reached by the time . This means that the oscillation amplitude of these harmonics strongly depends on , as it was shown theoretically, see the second line of Eq. (28). It can be concluded that the theory of thin shells describes very well the dynamics of CNTs under pulsed pressure. This conclusion is supported by comparison of the critical pressure values calculated using molecular dynamics and the shell theory, see Fig. 4(b), and also by the dynamics of various harmonics under constant and pulsed pressure, see Fig. 5 and Fig. 7. Carbon Nanotube under Pulsed Pressure 289 Fig. 7 (a) Amplitudes of Fourier harmonics as the functions of time found for CNT with N=200 under external pulsed pressure (a) p=18.2 MPa (=0.816), (b) p=36.4 MPa (=2.0), and (c) p=54.6 MPa (=3.0). Pressure drops to zero at time  shown by the vertical dashed line: in (a) =100 ps, (b) =25 ps, and (c) =20 ps. 5. CONCLUSIONS At present, the static deformation of the CNT cross section under the action of external pressure has been studied and a strong dependence on the CNT diameter has been shown. The spectral properties of the nanotubes were also analyzed depending on the external pressure. To the best of our knowledge, the dynamic behavior of CNTs under the action of a pressure pulse of various durations and intensities has been less studied. In this work, to analyze the dynamics of CNTs under plane strain conditions, we applied a continuum approach based on the theory of thin cylindrical shells based on the Kirchhoff hypotheses. In addition, we used the molecular dynamics method based on the chain model, which describes the deformation of the CNT cross section under a uniform external pressure. By comparing the eigenfrequencies of radial vibrations of a circular ring and CNT wall consisting of a different number of atoms N , the parameter  is determined, which contains the bending stiffness and density of the ring, see Table 1. 290 M. A. ILGAMOV, A. A. AITBAEVA, I. S. PAVLOV, S. V. DMITRIEV The introduction of the parameter  into the analysis opens up the possibility of a wide application of the models and relations of the well-developed theory of elastic thin shells. In this case, it is not necessary to involve such parameters as effective density, elastic modulus, Poisson’s ratio, wall thickness, the determination of which is a difficult task. Determining the parameter  from such an integral characteristic as natural frequencies of CNT vibrations seems to be more justified than from the effective parameters, which are given in the literature with a wide spread. For example, the effective thickness of the CNT wall is given in the literature in the range 0.7-3.4 Å. From the analysis of the values of the parameter  for CNTs of different diameters, determined by the chirality index n (or number of carbon atoms in a CNT cross section N=2n) given in Table. 1, one can conclude that for the zigzag CNTs (n,0) with n>15, in the regime of small amplitude vibrations, the discrepancy between the continuum model and molecular dynamics calculations does not exceed 5% and decreases with increasing n. As a result of the theoretical analysis carried out within the framework of the linear theory of thin shells, the dynamics of the deflection function of the CNT wall is described under the instantaneous application of a time constant or pulsed external pressure. Theoretical results are compared with molecular dynamics calculations, where the expansion of the deflection function in a Fourier series was analyzed. In particular, it is shown that the dynamics of bending harmonics with numbers m≥2 changes qualitatively when the external pressure exceeds the critical values pm * determined from Eq. (18). At pressure in the range p* m < p < p* m+1, the amplitude of harmonics with numbers less than m+1 grows exponentially, while harmonics with higher numbers exhibit oscillatory motion. Interestingly, the fastest increase in the amplitude is observed for the harmonic at the border of exponential growth and oscillatory motion. Under the action of a rectangular pressure pulse of short duration (compared to the oscillation period of the fundamental bending harmonic), the dynamics of different modes depends both on the value of the applied pressure p and on the pulse duration . 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